Finite Group Databasebeta

← #10608264

Subgroups — #10608264

Order 640.

704 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4093C211yesmaximal, characteristic, complemented
2C2×#4093C211yesmaximal, characteristic, complemented
3C2×#4039C211yesmaximal, characteristic, complemented
4C2×#4020C211yesmaximal, characteristic
5C5×#391C211yesmaximal, characteristic, complemented
6#65277C211yesmaximal, characteristic, complemented
7C4×#683C211yesmaximal, characteristic, complemented
8#64961C214yesmaximal, complemented
9#64954C214yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×D20C2211yescharacteristic
12C22×D20C2214yes—
13C22×D20C2211yescharacteristic
14C22×C4×D5C2212yes—
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C42×C5C2211yescharacteristic
17#4093—24no—
18#4093—24no—
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×#685C2214yes—
22C2×#685C2212yes—
23C2×#685C2214yes—
24C2×#683C2212yes—
25C2×C4×Dic5C2211yescharacteristic
26C2×C4×Dic5C2211yescharacteristic
27#4039—24no—
28#4020—24no—
29C5×#95C2214yes—
30#3963C2212yes—
31#2544—51nomaximal, Sylow
32C23×D5C2312yes—
33C23×D5C2312yes—
34C2×C5×Q8—22no—
35C2×C5×Q8—22no—
36C2×C5×Q8—22no—
37C22×C4×C5C2312yes—
38C22×C4×C5C2312yes—
39C22×C4×C5C2312yes—
40C22×C4×C5C2311yescharacteristic
41C22×Dic5C2312yes—
42C22×Dic5C2311yescharacteristic
43C22×Dic5C2311yescharacteristic
44C2×D20—22no—
45C2×D20—22no—
46C2×D20—28no—
47C2×D20—22no—
48C2×D20—22no—
49C2×D20—22no—
50C2×D20—28no—
51C2×D20—44no—
52C2×D20D414yes—
53C2×C4×D5—24no—
54C2×C4×D5—44no—
55C5×C4⋊C4—22no—
56C42×C5—22no—
57#685—28no—
58#685—24no—
59#685—24no—
60#685—24no—
61#685—44no—
62#683—24no—
63C4×Dic5—22no—
64C4×Dic5D414yescomplemented
65C2×#127—51no—
66C2×#124—51no—
67C2×#124—51no—
68C2×C4×D4—51no—
69#394—54no—
70#391D511yescharacteristic, complemented
71#390—51no—
72#386—54no—
73C4×C4⋊C4—51no—
74C23×C5C2411yescharacteristic
75C22×D5—22no—
76C22×D5—22no—
77C22×D5—24no—
78C22×D5—24no—
79C22×D5—22no—
80C22×D5—22no—
81C22×D5—22no—
82C22×D5—24no—
83C22×D5—24no—
84C22×D5C2×D414yes—
85C5×Q8—44no—
86C2×C4×C5—22no—
87C2×C4×C5—22no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—24no—
93C2×C4×C5C2×D412yes—
94C2×C4×C5C2×D412yes—
95C2×C4×C5#4112yes—
96C2×C4×C5#4114yes—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5C2×D412yes—
103C2×Dic5C2×D412yes—
104D20—28no—
105D20—416no—
106D20—44no—
107C4×D5—48no—
108C22×Q8C2×D511yescharacteristic
109C22×D4—51no—
110C22×D4—51no—
111C22×D4—54no—
112C23×C4—52no—
113#127—104no—
114#124—104no—
115#124—104no—
116C4×D4—104no—
117C2×C4⋊C4—52no—
118C2×C4⋊C4C2×D511yescharacteristic
119C2×C22⋊C4—52no—
120C2×C22⋊C4—52no—
121C2×C22⋊C4—54no—
122C2×C22⋊C4—52no—
123C2×C22⋊C4—54no—
124C2×C42—51no—
125C2×C42—51no—
126C2×C42C2×D511yescharacteristic
127#95—52no—
128#95C2×D514yes—
129C22×C5C22×D411yescharacteristic
130C22×C5C22×D411yescharacteristic
131C22×C5C2×#4111yescharacteristic
132C22×C5C2×#4112yes—
133C22×C5H(5,𝔽2)12yes—
134C2×D5—24no—
135C2×D5—28no—
136C2×D5—42no—
137C2×D5—42no—
138C2×D5—44no—
139C2×D5—42no—
140C2×D5—42no—
141C2×D5—42no—
142C2×D5—44no—
143C2×D5—44no—
144C4×C5—24no—
145C4×C5—42no—
146C4×C5—42no—
147C4×C5#12114yes—
148Dic5—22no—
149Dic5—22no—
150Dic5—42no—
151C24—52no—
152C24—52no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×D4—54no—
157C2×D4—108no—
158C2×D4—102no—
159C2×D4—102no—
160C2×D4—102no—
161C2×D4—102no—
162C2×D4—102no—
163C2×D4—108no—
164C2×D4—204no—
165C22×C4—51no—
166C22×C4—51no—
167C22×C4—52no—
168C22×C4—104no—
169C22×C4—204no—
170C22×C4C22×D512yes—
171C22×C4C22×D512yes—
172C22×C4C22×D512yes—
173C22×C4C22×D511yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—104no—
176C22⋊C4—104no—
177C22⋊C4—104no—
178C22⋊C4—104no—
179C22⋊C4—108no—
180C22⋊C4—204no—
181C42—22no—
182C42—54no—
183C42—102no—
184C2×C5C2×#12111yescharacteristic
185C2×C5#53211yescharacteristic
186C2×C5#53511yescharacteristic
187C2×C5#54612yes—
188C2×C5#55912yes—
189D5—44no—
190D5—82no—
191C23—54no—
192C23—102no—
193C23—102no—
194C23—104no—
195C23—104no—
196C23—102no—
197C23—102no—
198C23—102no—
199C23—104no—
200C23—104no—
201C23C23×D511yescharacteristic
202Q8—44no—
203D4—108no—
204D4—2016no—
205D4—204no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—52no—
214C2×C4—52no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—208no—
221C2×C4#70912yes—
222C2×C4D4×D512yes—
223C2×C4D4×D512yes—
224C2×C4#71314yes—
225C5#254411yescharacteristic, complemented, Sylow
226C22—104no—
227C22—108no—
228C22—202no—
229C22—204no—
230C22—202no—
231C22—202no—
232C22—202no—
233C22—204no—
234C22—204no—
235C22—202no—
236C22C2×#70911yescharacteristic
237C22C2×D4×D511yescharacteristic
238C22C2×D4×D511yescharacteristic
239C22C2×#71312yes—
240C22#414912yes—
241C4—24no—
242C4—42no—
243C4—42no—
244C4—102no—
245C4—102no—
246C4—202no—
247C4#403014yes—
248C2—204no—
249C2—402no—
250C2C2×#403011yescharacteristic
251C2#6591912yes—
252C2#6596412yes—
253C2#6600811yescharacteristic
254C2#6605911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#391C2
prime radicalC2×#4020C2
hypercenter#391D5
hypocenterC5#2544
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #391

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#4093C2semidirect
2. C2×#4093C2semidirect
3. C2×#4039C2semidirect
5. C5×#391C2semidirect
6. #65277C2semidirect
7. C4×#683C2semidirect
8. #64961C2semidirect
9. #64954C2semidirect
64. C4×Dic5D4semidirect
70. #391D5semidirect
225. C5#2544semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#40932
C2×#40392
C2×#40202
C5×#3912
#652772
C4×#6832
#649612
#649542
#25445